A good easy trip for you would be 9 Mile canyon. The roads are graded and there are petroglyphs and runs right on the road or a short hike from it. You can stay in Price or Vernal. The start of that drive is about 2 hours outside of Salt Lake. 9 Mile canyon is actually about 45 miles long. Here's a link to a visitors guide.
Nine Mile Canyon Map and Guide - Castle Country Utah
Start in SLC and head to Myton, UT. (2 HRS 15 Minutes on the highway)
Do the 9 Mile canyon drive, stay in Price.
From Price head South. 2 great options:
1. The wedge overlook:
Wedge Overlook and Buckhorn Draw Backway
Head West at Wellington towards Cleveland. A decent map will be your friend to get through the back roads. This is a graded road also, but there are some 4x4 roads that break off the main road. It's remote, but not so remote you're in danger when going solo. Head out Buckhorn Draw road to drive through high walled canyons with red cliffs and see the MK tunnels and more petroglyphs. For this route stay in Green River at a hotel after your day on the road.
2. If you have 3 days you can do this after the wedge overlook. Head South from Green River on HWY 24 towards Hanksville. To to Goblin Valley State Park. It's worth the $10 entrance fee (maybe $15). Spend an hour or two with a light hike through the goblins. Leave the park and go North. At the point you would turn right to get back to the highway, go left and drive on the behind the reef road. There are 2 easy narrow canyons not too far in: Crack Canyon and Chute Canyon. The other slot canyons require some scrambling. These two have a couple places you have to climb up or down, but they're not too bad. If you don't want to hike this road continue for several (15?) Miles, out and back. Stay in Green River again or head back to Price. It's about 4 hours to Salt Lake from here.
So far all of these routes are passable with a high clearance 2WD vehicle. Yours will be fine. For more driving adventure there's a network of roads that go from I-70 to Goblin Valley that may present more of the challenge you are looking for, but you may or may not see another car out there.
If you want to do the wedge overlook head down to Capitol Reef. There's a ton to see down there. Upper Muley Twist on the Burr trail is a good drive and will exercise your GX a little, but you'll make it. Take the short overlook hike for spectacular views. Don't miss the double watch in the canyon. There are hotels in the highway West of Capitol Reef. You can get to them on hwy 12 after you've driven the scenic Burr trail or stay there and explore the park from the north end. It's about an hour to drive to the Burr trail from the north end.
Head down hwy 12 and you have all of the Escalante National Monument to explore. Now you're 6 hours from salt lake. Hotels in Escalante, or head north and stop somewhere on the way back.
-Hells backbone
-Calf Creek falls
-Hole in the rock road
